This invention pertains to the field of topology, and more particularly to a method for restoring the continuity of a property interpolated over a modeled surface across discontinuities that can affect this surface.
One method in the prior art for modeling a two dimensional surface which varies in a third dimension is to generate a topological surface composed of a plurality of triangles having apexes connected at nodes. The surfaces themselves are modeled as piece-wise continuous sets of triangles. Such a surface is commonly termed a triangulated surface. In modeling such surfaces, the surface typically has a property applied to or mapped to the surface. The property can be as simple as the variation of nodes of the triangles from an initial reference point, or variation from the adjacent node. Alternately, the surface can be described by a physical property along the surface, such as temperature. Typically, both the geometry and the properties are discrete, and are stored only at the node locations.
Examples of surfaces which can be modeled in this way are a contoured surface such as a body panel for a vehicle; interfaces between geological strata, interfaces between organic tissue; and even the surface of an animated figure, to name but a few. Often such surfaces have discontinuities. For example, a body panel for a vehicle can have an opening for a fuel fill nozzle, or a geological strata can be separated along a fault line. In modeling such surfaces, it is desirable that the property modeled across such discontinuities in the surface be modeled such that the property accurately fits the surface.
Current methods for modeling a property across a discontinuity in a surface, such as a triangluated surface, produce results which are not typically representative of the property. For example, a linear interpolation is assumed to establish the geometry and properties inside each triangle. However, current methods for interpolating properties on such a surface leads to a clearly discontinuous result. Referring now to FIG. 1, a diagram showing a property interpolated over a discontinuous surface S having discontinuities D1, D2 and D3 is shown. The property depicted is interpolated using prior art methods. The property, displayed here as contour lines L, is also discontinuous, as can be seen at points P1 and P2 wherein the line L1 does not smoothly interpolate across the discontinuity D1. Further, the lines inside discontinuities D1 and D3 represent the directions along which it is preferred to reconcile the two sides of the discontinuity when interpolating this property.
It is therefore desirable to have a method for smoothly interpolating a property across a discontinuity in a modeled surface, especially in a triangulated surface.
The present invention includes a method for restoring the continuity of a property interpolated over a surface and across discontinuities that can affect the same surface.
One aspect of the present invention includes a method of establishing a continuity of a property which is interpolated over a surface. In this regard the method includes the steps of identifying first and second portions of a boundary which defines a discontinuity in the surface, the surface defined by at least two primary nodes; defining the first and second portions of the boundary by at least one secondary node on each portion; and interpolating the property from a first primary node to a second primary node by utilizing the secondary nodes.
Another aspect of the present invention relates to a method for modeling a value of a property across a discontinuity which is defined in a surface. The method comprises the steps of defining the surface and first and second portions of a boundary of a discontinuity by a plurality of primary nodes; assigning a secondary node to each of the first and second portions of the boundary; and interpolating the value of the property across the boundary from a first primary node to a second primary node while maintaining a predetermined continuity between the corresponding secondary nodes.
A further aspect of the present invention includes a method of establishing a continuity of a property which is interpolated over a surface. The method comprises the following sequential steps: first, identifying first and second portions of a boundary which defines a discontinuity in the surface, the surface defined by at lest two primary nodes. Following identifying the first and second portions of the boundary, the method comprises the step of defining the first and second portions of the boundary by at least one secondary node on each portion of the boundary. Following the step of defining the first and second portions of the boundary by at least one secondary node on each portion, the method further comprises the step of substantially simultaneously interpolating the property from a first primary node to a second primary node by utilizing the secondary nodes.
An additional aspect of the present invention includes a method of calculating values for a property to be mapped onto a surface to provide a continuity of values of the property across a discontinuity which is defined by the surface. The method comprises topologically defining the surface by a plurality of primary nodes which are mathematically associated with one another to provide a plurality of triangles having apexes at the primary nodes and triangle edges connected the primary nodes. Thereafter, the method comprises identifying first and second boundaries which define the discontinuity in the surface, the boundaries comprising triangle edges and identifying selected primary nodes which are on the first and second boundaries of the discontinuity. The method also comprises segmenting the boundaries of the discontinuity into an equal number of segments, and identifying locations of segments along the triangle edges which comprise the first and second boundaries of the discontinuity. In this step, a first value of the property is calculated for a first primary node of the second continuity, and a second value of the property is calculated for a second primary node on the first boundary of the discontinuity. The method also comprises linearly interpolating a third value of the property at the location of a segment on the first boundary as a function of the distance of the segment from the first primary node, and also as a function of the first and second values of the property at the respective first and second primary nodes. The third value of the property at the location of the segment on the first boundary is assigned to the corresponding location of the segment on the second boundary. The method further comprises a step of calculating a fourth value of the property at a third primary node on the second boundary as a function of the distance of the third primary node from the location of the segment on the second boundary and as a function of the third value of the property.